Natural number
, two apples, three apples, ...)]] In mathematics, the natural numbers (sometimes called the whole numbers) : "whole number An integer, though sometimes it is taken to mean only non-negative integers, or just the positive integers." give definitions of "whole number" under several headwords: INTEGER … Syn. whole number. NUMBER … whole number. A nonnegative integer. WHOLE … whole number. (1) One of the integers 0, 1, 2, 3, … . (2) A positive integer; i.e., a natural number. (3) An integer, positive, negative, or zero. The Common Core State Standards for Mathematics say: "Whole numbers. The numbers 0, 1, 2, 3, ...." (Glossary, p. 87) (PDF) Definitions from The Ontario Curriculum, Grades 1-8: Mathematics, Ontario Ministry of Education (2005) (PDF) "natural numbers. The counting numbers 1, 2, 3, 4, ...." (Glossary, p. 128) "whole number. Any one of the numbers 0, 1, 2, 3, 4, ...." (Glossary, p. 134) : "As mentioned earlier, the study of the set of whole numbers, W'' = {0, 1, 2, 3, 4, ...}, is the foundation of elementary school mathematics." These pre-algebra books define the ''whole numbers: * : "Another important collection of numbers is the whole numbers, the natural numbers together with zero." (Chapter 1: The Whole Story, p. 4). On the inside front cover, the authors say: "We based this book on the state standards for pre-algebra in California, Florida, New York, and Texas, ..." * : "When 0 is added to the set of natural numbers, the set is called the whole numbers." (Chapter 1: Whole Numbers, p. 1) :Both books define the natural numbers to be: "1, 2, 3, …". are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). In common language, words used for counting are "cardinal numbers" and words used for ordering are "ordinal numbers". Another use of natural numbers is for what linguists call nominal numbers, such as the model number of a product, where the "natural number" is used only for naming (as distinct from a serial number where the order properties of the natural numbers distinguish later uses from earlier uses) and generally lacks any meaning of number as used in mathematics. The natural numbers are the basis from which many other number sets may be built by extension: the integers, by including an unresolved negation operation; the rational numbers, by including with the integers an unresolved division operation; the real numbers by including with the rationals the termination of Cauchy sequences; the complex numbers, by including with the real numbers the unresolved square root of minus one; the hyperreal numbers, by including with real numbers the infinitesimal value epsilon; vectors, by including a vector structure with reals; matrices, by having vectors of vectors; the nonstandard integers; and so on. says: "The whole fantastic hierarchy of number systems is built up by purely set-theoretic means from a few simple assumptions about natural numbers." (Preface, p. x) : "Numbers make up the foundation of mathematics." (p. 1) Therefore, the natural numbers are canonically embedded (identified) in the other number systems. Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. There is no universal agreement about whether to include zero in the set of natural numbers. Some authors begin the natural numbers with , corresponding to the non-negative integers , whereas others start with 1, corresponding to the positive integers , , , ...}}. says: "ℕ is the set of natural numbers (positive integers)" (p. 3) include zero in the natural numbers: 'Intuitively, the set ℕ = {0, 1, 2, ... } of all natural numbers may be described as follows: ℕ contains an "initial" number 0; ...'. They follow that with their version of the Peano Postulates. (p. 15) This distinction is of no fundamental concern for the natural numbers as such, since their core construction is the unary operation successor. Including the number 0 just supplies an identity element for the (binary) operation of addition, which makes up together with the multiplication the usual arithmetic in the natural numbers, to be completed within the integers and the rational numbers, only. In common language, for example in primary school, natural numbers may be called counting numbers to distinguish them from the real numbers which are used for measurement. References